Modeling of nonlinear deformation of a plate with elliptic inclusion by John’s harmonic material

Abstract

The exact analytical solution of nonlinear plane-strain problems for a plate with an elastic elliptic inclusion is obtained. The constant stresses are given at infinity. The mechanical properties of a plate and inclusion are described by model of a John’s harmonic material. For this model the methods of complex functions are applicable to solution of nonlinear plane problems. It is supposed, that the state of stress inside inclusion is homogeneous (tensor of nominal stresses is constant). This assumption has allowed to reduce a difficult nonlinear problem of elliptic inclusion in plate to the solution of two more simple problems for a plate with an elliptic hole. A validity of adopted hypothesis is justified by that the received solution precisely satisfies to all equations and boundary conditions of a problem. It is established, that material can lose stability at uniaxial or biaxial compression of a plate. The solutions of some partial nonlinear problems are received from the common solution: a plate with a free elliptic hole and a plate with rigid inclusion. Refs 13. Figs 3.